Bartlett分解定理的简便证明

A Concise Proof of the Bartlett Decomposition Theorem

对服从Wishart分布的随机矩阵W～Wp(n ,Ι)已有著名的Bartlett分解定理 ,结果非常完美 ,但证明过程既繁又长 ,本文用特征函数方法证明 2个服从n -i+ 1维标准正态分布、且相互独立的随机向量的内积应同分布于一个服从 χn -i + 1分布的随机变量与一个与其独立且服从N( 0 ,1 )分布的随机变量的乘积 .从而简单而直观地证明该定理 ,虽结论稍减弱为W =d T′T 。

For a random matrix that represents the Whishart distribution, there exists the well-known Bartlett decomposition theorem. The result is perfect, but the proof is tedious. This article uses the characteristic function to prove the following result: the inner product of two independent (n-i+1) -dimensional random vectors subject to standard normal distribution has the same distribution of the product of two independent random variables in which one is subject to χ n-i+1 distribution and the other to N(0,1) distribution. Thus the theorem can be proved briefly and directly. Though the conclusion is reduced to W= d T′T , it does not affect the application of Bartlett decomposition theorem in most cases.